Typesetting: Photocomposed pages prepared by the authors using plain TeX files.www.springer-ny.comSpringer-Verlag New York Berlin HeidelbergA member of BertelsmannSpringer Science+Business Media GmbH

◆To Nola,and in memory of Mary◆

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Preface to the Second Edition

In preparing this second edition, we have taken the opportunity to reshapethe book, partly in response to the further explosion of material on pointprocesses that has occurred in the last decade but partly also in the hopeof making some of the material in later chapters of the ﬁrst edition moreaccessible to readers primarily interested in models and applications. Topicssuch as conditional intensities and spatial processes, which appeared relativelyadvanced and technically diﬃcult at the time of the ﬁrst edition, have nowbeen so extensively used and developed that they warrant inclusion in theearlier introductory part of the text. Although the original aim of the book—to present an introduction to the theory in as broad a manner as we areable—has remained unchanged, it now seems to us best accomplished in twovolumes, the ﬁrst concentrating on introductory material and models and thesecond on structure and general theory. The major revisions in this volume,as well as the main new material, are to be found in Chapters 6–8. The restof the book has been revised to take these changes into account, to correcterrors in the ﬁrst edition, and to bring in a range of new ideas and examples.Even at the time of the ﬁrst edition, we were struggling to do justice tothe variety of directions, applications and links with other material that thetheory of point processes had acquired. The situation now is a great dealmore daunting. The mathematical ideas, particularly the links to statisticalmechanics and with regard to inference for point processes, have extendedconsiderably. Simulation and related computational methods have developedeven more rapidly, transforming the range and nature of the problems underactive investigation and development. Applications to spatial point patterns,especially in connection with image analysis but also in many other scientiﬁc disciplines, have also exploded, frequently acquiring special language andtechniques in the diﬀerent ﬁelds of application. Marked point processes, whichwere clamouring for greater attention even at the time of the ﬁrst edition, haveacquired a central position in many of these new applications, inﬂuencing boththe direction of growth and the centre of gravity of the theory.vii

viii

Preface to the Second Edition

We are sadly conscious of our inability to do justice to this wealth of newmaterial. Even less than at the time of the ﬁrst edition can the book claim toprovide a comprehensive, up-to-the-minute treatment of the subject. Nor arewe able to provide more than a sketch of how the ideas of the subject haveevolved. Nevertheless, we hope that the attempt to provide an introductionto the main lines of development, backed by a succinct yet rigorous treatmentof the theory, will prove of value to readers in both theoretical and appliedﬁelds and a possible starting point for the development of lecture courses ondiﬀerent facets of the subject. As with the ﬁrst edition, we have endeavouredto make the material as self-contained as possible, with references to background mathematical concepts summarized in the appendices, which appearin this edition at the end of Volume I.We would like to express our gratitude to the readers who drew our attention to some of the major errors and omissions of the ﬁrst edition andwill be glad to receive similar notice of those that remain or have been newlyintroduced. Space precludes our listing these many helpers, but we would liketo acknowledge our indebtedness to Rick Schoenberg, Robin Milne, VolkerSchmidt, G¨unter Last, Peter Glynn, Olav Kallenberg, Martin Kalinke, JimPitman, Tim Brown and Steve Evans for particular comments and carefulreading of the original or revised texts (or both). Finally, it is a pleasure tothank John Kimmel of Springer-Verlag for his patience and encouragement,and especially Eileen Dallwitz for undertaking the painful task of rekeying thetext of the ﬁrst edition.The support of our two universities has been as unﬂagging for this endeavour as for the ﬁrst edition; we would add thanks to host institutions of visitsto the Technical University of Munich (supported by a Humboldt FoundationAward), University College London (supported by a grant from the Engineering and Physical Sciences Research Council) and the Institute of Mathematicsand its Applications at the University of Minnesota.Daryl DaleyCanberra, Australia

David Vere-JonesWellington, New Zealand

Preface to the First Edition

This book has developed over many years—too many, as our colleagues andfamilies would doubtless aver. It was conceived as a sequel to the review paperthat we wrote for the Point Process Conference organized by Peter Lewis in1971. Since that time the subject has kept running away from us faster thanwe could organize our attempts to set it down on paper. The last two decadeshave seen the rise and rapid development of martingale methods, the surgeof interest in stochastic geometry following Rollo Davidson’s work, and theforging of close links between point processes and equilibrium problems instatistical mechanics.Our intention at the beginning was to write a text that would provide asurvey of point process theory accessible to beginning graduate students andworkers in applied ﬁelds. With this in mind we adopted a partly historicalapproach, starting with an informal introduction followed by a more detaileddiscussion of the most familiar and important examples, and then movinggradually into topics of increased abstraction and generality. This is still thebasic pattern of the book. Chapters 1–4 provide historical background andtreat fundamental special cases (Poisson processes, stationary processes onthe line, and renewal processes). Chapter 5, on ﬁnite point processes, has abridging character, while Chapters 6–14 develop aspects of the general theory.The main diﬃculty we had with this approach was to decide when andhow far to introduce the abstract concepts of functional analysis. With someregret, we ﬁnally decided that it was idle to pretend that a general treatment ofpoint processes could be developed without this background, mainly becausethe problems of existence and convergence lead inexorably to the theory ofmeasures on metric spaces. This being so, one might as well take advantageof the metric space framework from the outset and let the point process itselfbe deﬁned on a space of this character: at least this obviates the tedium ofhaving continually to specify the dimensions of the Euclidean space, while inthe context of completely separable metric spaces—and this is the greatestix

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Preface to the First Edition

generality we contemplate—intuitive spatial notions still provide a reasonableguide to basic properties. For these reasons the general results from Chapter6 onward are couched in the language of this setting, although the examplescontinue to be drawn mainly from the one- or two-dimensional Euclideanspaces R1 and R2 . Two appendices collect together the main results we needfrom measure theory and the theory of measures on metric spaces. We hopethat their inclusion will help to make the book more readily usable by appliedworkers who wish to understand the main ideas of the general theory withoutthemselves becoming experts in these ﬁelds. Chapter 13, on the martingaleapproach, is a special case. Here the context is again the real line, but weadded a third appendix that attempts to summarize the main ideas neededfrom martingale theory and the general theory of processes. Such specialtreatment seems to us warranted by the exceptional importance of these ideasin handling the problems of inference for point processes.In style, our guiding star has been the texts of Feller, however many lightyears we may be from achieving that goal. In particular, we have tried tofollow his format of motivating and illustrating the general theory with arange of examples, sometimes didactical in character, but more often takenfrom real applications of importance. In this sense we have tried to strikea mean between the rigorous, abstract treatments of texts such as those byMatthes, Kerstan and Mecke (1974/1978/1982) and Kallenberg (1975, 1983),and practically motivated but informal treatments such as Cox and Lewis(1966) and Cox and Isham (1980).Numbering Conventions. Each chapter is divided into sections, with consecutive labelling within each of equations, statements (encompassing Deﬁnitions, Conditions, Lemmas, Propositions, Theorems), examples, and the exercises collected at the end of each section. Thus, in Section 1.2, (1.2.3) is thethird equation, Statement 1.2.III is the third statement, Example 1.2(c)is the third example, and Exercise 1.2.3 is the third exercise. The exercisesare varied in both content and intention and form a signiﬁcant part of thetext. Usually, they indicate extensions or applications (or both) of the theoryand examples developed in the main text, elaborated by hints or referencesintended to help the reader seeking to make use of them. The symbol denotes the end of a proof. Instead of a name index, the listed references carrypage number(s) where they are cited. A general outline of the notation usedhas been included before the main text.It remains to acknowledge our indebtedness to many persons and institutions. Any reader familiar with the development of point process theory overthe last two decades will have no diﬃculty in appreciating our dependence onthe fundamental monographs already noted by Matthes, Kerstan and Meckein its three editions (our use of the abbreviation MKM for the 1978 Englishedition is as much a mark of respect as convenience) and Kallenberg in itstwo editions. We have been very conscious of their generous interest in oureﬀorts from the outset and are grateful to Olav Kallenberg in particular forsaving us from some major blunders. A number of other colleagues, notably

Preface to the First Edition

xi

David Brillinger, David Cox, Klaus Krickeberg, Robin Milne, Dietrich Stoyan,Mark Westcott, and Deng Yonglu, have also provided valuable comments andadvice for which we are very grateful. Our two universities have respondedgenerously with seemingly unending streams of requests to visit one anotherat various stages during more intensive periods of writing the manuscript. Wealso note visits to the University of California at Berkeley, to the Center forStochastic Processes at the University of North Carolina at Chapel Hill, andto Zhongshan University at Guangzhou. For secretarial assistance we wishto thank particularly Beryl Cranston, Sue Watson, June Wilson, Ann Milligan, and Shelley Carlyle for their excellent and painstaking typing of diﬃcultmanuscript.Finally, we must acknowledge the long-enduring support of our families,and especially our wives, throughout: they are not alone in welcoming thespeed and eﬃciency of Springer-Verlag in completing this project.Daryl DaleyCanberra, Australia

David Vere-JonesWellington, New Zealand

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Contents

Preface to the Second EditionPreface to the First Edition

viiix

Principal NotationConcordance of Statements from the First Edition1

Early History

1

1.1 Life Tables and Renewal Theory1.2 Counting Problems1.3 Some More Recent Developments2

Basic Properties of the Poisson Process

2.1 The Stationary Poisson Process2.2 Characterizations of the Stationary Poisson Process:I. Complete Randomness2.3 Characterizations of the Stationary Poisson Process:II. The Form of the Distribution2.4 The General Poisson Process3

Simple Results for Stationary Point Processes on the Line

3.13.23.33.43.53.6

xviixxi

Speciﬁcation of a Point Process on the LineStationarity: DeﬁnitionsMean Density, Intensity, and Batch-Size DistributionPalm–Khinchin EquationsErgodicity and an Elementary Renewal Theorem AnalogueSubadditive and Superadditive Functionsxiii

Likelihoods and Janossy DensitiesConditional Intensities, Likelihoods, and CompensatorsConditional Intensities for Marked Point ProcessesRandom Time Change and a Goodness-of-Fit TestSimulation and Prediction AlgorithmsInformation Gain and Probability Forecasts

Set TheoryTopologiesFinitely and Countably Additive Set FunctionsMeasurable Functions and IntegralsProduct SpacesDissecting Systems and Atomic MeasuresMeasures on Metric Spaces

A2.1A2.2A2.3A2.4A2.5A2.6A2.3A2.3A3A3.1A3.2A3.3A3.4

xv

368368369372374377382384

Borel Sets and the Support of MeasuresRegular and Tight MeasuresWeak Convergence of MeasuresCompactness Criteria for Weak ConvergenceMetric Properties of the Space MXBoundedly Finite Measures and the Space M#XMeasures on Topological GroupsFourier Transforms

Very little of the general notation used in Appendices 1–3 is given below. Also,notation that is largely conﬁned to one or two sections of the same chapteris mostly excluded, so that neither all the symbols used nor all the uses ofthe symbols shown are given. The repeated use of some symbols occurs as aresult of point process theory embracing a variety of topics from the theory ofstochastic processes. Where they are given, page numbers indicate the ﬁrstor signiﬁcant use of the notation. Generally, the particular interpretation ofsymbols with more than one use is clear from the context.Throughout the lists below, N denotes a point process and ξ denotes arandom measure.

The table below lists the identifying number of formal statements of the ﬁrstedition (1988) of this book and their identiﬁcation in this volume.1988 edition

this volume

1988 edition

this volume

2.2.I–III

2.2.I–III

2.3.III2.4.I–II2.4.V–VIII

2.3.I2.4.I–II2.4.III–VI

3.2.I–II3.3.I–IX

3.2.I–II3.3.I–IX

8.1.II8.2.I8.2.II8.3.I–III8.5.I–III

6.1.II, IV6.3.I6.3.II, (6.3.6)6.3.III–V6.2.II

11.1.I–V

8.6.I–V

3.4.I–II3.5.I–III3.6.I–V

3.4.I–II3.5.I–III3.6.I–V

11.2.I–II11.3.I–VIII

8.2.I–II8.4.I–VIII

4.2.I–II4.3.I–III4.4.I–VI4.5.I–VI

4.2.I–II4.3.I–III4.4.I–VI4.5.I–VI

11.4.I–IV11.4.V–VI

8.5.I–IV8.5.VI–VII

13.1.I–III13.1.IV–VI13.1.VII

7.1.I–III7.2.I–III7.1.IV

13.4.III

7.6.I

4.6.I–V

4.6.I–V

5.2.I–VII5.3.I–III5.4.I–III5.4.IV–VI5.5.I

5.2.I–VII5.3.I–III5.4.I–III5.4.V–VII5.5.I

A1.1.I–5.IVA2.1.I–IIIA2.1.IVA2.1.V–VIA2.2.I–7.IIIA3.1.I–4.IX

A1.1.I–5.IVA2.1.I–IIIA1.6.IA2.1.IV–VA2.2.I–7.IIIA3.1.I–4.IX

7.1.XII–XIII

6.4.I(a)–(b)xxi

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CHAPTER 1

Early History

The ancient origins of the modern theory of point processes are not easy totrace, nor is it our aim to give here an account with claims to being deﬁnitive.But any retrospective survey of a subject must inevitably give some focus onthose past activities that can be seen to embody concepts in common with themodern theory. Accordingly, this ﬁrst chapter is a historical indulgence butwith the added beneﬁt of describing certain fundamental concepts informallyand in a heuristic fashion prior to possibly obscuring them with a plethora ofmathematical jargon and techniques. These essentially simple ideas appearto have emerged from four distinguishable strands of enquiry—although ourdivision of material may sometimes be a little arbitrary. These are(i)(ii)(iii)(iv)

life tables and the theory of self-renewing aggregates;counting problems;particle physics and population processes; andcommunication engineering.

The ﬁrst two of these strands could have been discerned in centuries pastand are discussed in the ﬁrst two sections. The remaining two essentiallybelong to the twentieth century, and our comments are briefer in the remainingsection.

1.1. Life Tables and Renewal TheoryOf all the threads that are woven into the modern theory of point processes,the one with the longest history is that associated with intervals betweenevents. This includes, in particular, renewal theory, which could be deﬁnedin a narrow sense as the study of the sequence of intervals between successivereplacements of a component that is liable to failure and is replaced by a new1

2

1. Early History

component every time a failure occurs. As such, it is a subject that developed during the 1930s and reached a deﬁnitive stage with the work of Feller,Smith, and others in the period following World War II. But its roots extendback much further than this, through the study of ‘self-renewing aggregates’to problems of statistical demography, insurance, and mortality tables—inshort, to one of the founding impulses of probability theory itself. It is noteasy to point with conﬁdence to any intermediate stage in this chronicle thatrecommends itself as the natural starting point either of renewal theory or ofpoint process theory more generally. Accordingly, we start from the beginning, with a brief discussion of life tables themselves. The connection withpoint processes may seem distant at ﬁrst sight, but in fact the theory of lifetables provides not only the source of much current terminology but also thesetting for a range of problems concerning the evolution of populations intime and space, which, in their full complexity, are only now coming withinthe scope of current mathematical techniques.In its basic form, a life table consists of a list of the number of individuals,usually from an initial group of 1000 individuals so that the numbers areeﬀectively proportions, who survive to a given age in a given population.The most important parameters are the number x surviving to age x, thenumber dx dying between the ages x and x + 1 (dx = x − x+1 ), and thenumber qx of those surviving to age x who die before reaching age x + 1(qx = dx / x ). In practice, the tables are given for discrete ages, with theunit of time usually taken as 1 year. For our purposes, it is more appropriateto replace the discrete time parameter by a continuous one and to replacenumbers by probabilities for a single individual. Corresponding to x we havethen the survivor functionS(x) = Pr{lifetime > x}.To dx corresponds f (x), the density of the lifetime distribution function, wheref (x) dx = Pr{lifetime terminates between x and x + dx},while to qx corresponds q(x), the hazard function, whereq(x) dx = Pr{lifetime terminates between x and x + dx| it does not terminate before x.}Denoting the lifetime distribution function itself by F (x), we have the following important relations between the functions above:S(x) = 1 − F (x) =

∞

x

f (y) dy = expx

−

q(y) dy ,

dFdS=,dxdxddf (x)=[log S(x)] = − {log[1 − F (x)]}.q(x) =S(x)dxdx

f (x) =

(1.1.1)

0

(1.1.2)(1.1.3)

1.1.

Life Tables and Renewal Theory

3

The ﬁrst life table appeared, in a rather crude form, in John Graunt’s (1662)Observations on the London Bills of Mortality. This work is a landmark in theearly history of statistics, much as the famous correspondence between Pascaland Fermat, which took place in 1654 but was not published until 1679, isa landmark in the early history of formal probability. The coincidence indates lends weight to the thesis (see e.g. Maistrov, 1967) that mathematicalscholars studied games of chance not only for their own interest but for theopportunity they gave for clarifying the basic notions of chance, frequency, andexpectation, already actively in use in mortality, insurance, and populationmovement contexts.An improved life table was constructed in 1693 by the astronomer Halley,using data from the smaller city of Breslau, which was not subject to thesame problems of disease, immigration, and incomplete records with whichGraunt struggled in the London data. Graunt’s table was also discussed byHuyghens (1629–1695), to whom the notion of expected length of life is due.A. de Moivre (1667–1754) suggested that for human populations the functionS(x) could be taken to decrease with equal yearly decrements between the ages22 and 86. This corresponds to a uniform density over this period and a hazardfunction that increases to inﬁnity as x approaches 86. The analysis leadingto (1.1.1) and (1.1.2), with further elaborations to take into account diﬀerentsources of mortality, would appear to be due to Laplace (1747–1829). It isinteresting that in A Philosophical Essay on Probabilities (1814), where theclassical deﬁnition of probability based on equiprobable events is laid down,Laplace gave a discussion of mortality tables in terms of probabilities of atotally diﬀerent kind. Euler (1707–1783) also studied a variety of problems ofstatistical demography.From the mathematical point of view, the paradigm distribution functionfor lifetimes is the exponential function, which has a constant hazard independent of age: for x > 0, we havef (x) = λe−λx ,

q(x) = λ,

S(x) = e−λx ,

F (x) = 1 − e−λx .

(1.1.4)

The usefulness of this distribution, particularly as an approximation for purposes of interpolation, was stressed by Gompertz (1779–1865), who also suggested, as a closer approximation, the distribution function corresponding toa power-law hazard of the formq(x) = Aeαx

(A > 0, α > 0, x > 0).

(1.1.5)

With the addition of a further constant [i.e. q(x) = B + Aeαx ], this is knownin demography as the Gompertz–Makeham law and is possibly still the mostwidely used function for interpolating or graduating a life table.Other forms commonly used for modelling the lifetime distribution in different contexts are the Weibull, gamma, and log normal distributions, corresponding, respectively, to the formulaeq(x) = βλxβ−1